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Domenico Fiorenza, Elena Martinengo,
A short note on $\infty$-groupoids and the period map for projective manifolds, Publications of the nLab vol. 2 no. 1 (2012) arXiv:0911.3845
Dipartimento di Matematica - Sapienza, Università di Roma; P.le Aldo Moro 5, I-00185 Roma Italy - fiorenza@mat.uniroma1.it
Institut für Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany - elenamartinengo@gmail.com
We show how several classical results on the infinitesimal behaviour of the period map for smooth projective manifolds can be read in a natural and unified way within the framework of ∞-categories.
A common criticism of ∞-categories in algebraic geometry is that they are an extremely technical subject, so abstract to be useless in everyday mathematics. The aim of this note is to show in a classical example that quite the converse is true: even a naïve intuition of what an ∞-groupoid should be clarifies several aspects of the infinitesimal behaviour of the periods map of a projective manifold. In particular, the notion of Cartan homotopy turns out to be completely natural from this perspective, and so classical results such as Griffiths’ expression for the differential of the periods map, the Kodaira principle on obstructions to deformations of projective manifolds, the Bogomolov-Tian-Todorov theorem, and Goldman-Millson quasi-abelianity theorem are easily recovered.
The use of the language of ∞-categories should not be looked at as providing new proofs for these results; namely, up to a change in language, our proofs verbatim reproduce arguments from the recent literature on the subject, particularly from the work of Marco Manetti and collaborators on dglas in deformation theory. Rather, by this change of language we change our point of view on the classical theorems above: in the perspective of ∞-sheaves from Lu09a, all these theorems have a very simple local nature which can be naturally expressed in terms of ∞-groupoids (or, equivalently, of dglas); their classical global counterparts are then obtained by taking derived global sections. It is worth remarking that, if one prefers proofs which do not rely on the abstract machinery of ∞-categories, one can rework the arguments of this note in purely classical terms. Namely, once the abstract $\infty$-nonsense has suggested the “correct” local dglas, one can globalize them by means of an explicit model for the derived global sections, e.g., via resolutions by fine sheaves as in FM09, or by the Thom-Sullivan-Whitney model as in IM10.
Since most of the statements and constructions we recall in the paper are well known in the $(\infty ,1)$-categorical folklore, despite our efforts in giving credit, it is not unlikely we may have misattributed a few of the results; we sincerely apologize for this. We thank the referee for accurate remarks which helped us a lot in improving the present paper, and Ezra Getzler, Donatella Iacono, Marco Manetti, Jonathan Pridham, Carlos Simpson, Jim Stasheff, Bruno Vallette, Gabriele Vezzosi, and the $n$Lab for suggestions and several inspiring conversations on the subject of this paper.
Through the whole paper, $\mathbb{K}$ is a fixed characteristic zero field, all algebras are defined over $\mathbb{K}$ and local algebras have $\mathbb{K}$ as residue field. In order to keep our account readable, we will gloss over many details, particularly where the use of higher category theory is required.
With any nilpotent dgla $\mathfrak{g}$ is naturally associated the simplicial set
where $\MC$ stands for the Maurer-Cartan functor mapping a dgla to the set of its Maurer-Cartan elements, and $\Omega _{\bullet }$ is the simplicial differential graded commutative associative algebra of polynomial differential forms on algebraic $n$-simplexes, for $n\geq 0$. The importance of this construction, which can be dated back to Sullivan’s Su77, relies on the fact that, as shown by Hinich and Getzler Hi97, Ge09, the simplicial set $\MC(\mathfrak{g}\otimes \Omega _{\bullet })$ is a Kan complex, or -to use a more evocative name- an ∞-groupoid. A convenient way to think of ∞-groupoids is as homotopy types of topological spaces; namely, it is well known^{1} that any ∞-groupoid can be realized as the ∞-Poincaré groupoid, i.e., as the simplicial set of singular simplices, of a topological space, unique up to weak equivalence. Therefore, the reader who prefers to can substitute homotopy types of topological spaces for equivalence classes of ∞-groupoids. To stress this point of view, we’ll denote the $k$-truncation of an ∞-groupoid $\mathbf{X}$ by the symbol $\pi _{\leq k}\mathbf{X}$. More explicitely, $\pi _{\leq k}\mathbf{X}$ is the $k$-groupoid whose $j$-morphisms are the $j$-morphisms of $\mathbf{X}$ for $j\lt k$, and are homotopy classes of $j$-morphisms of $\mathbf{X}$ for $j=k$. In particular, if $\mathbf{X}$ is the ? spring groupoid of a topological space $X$, then $\pi _{\leq 0}\mathbf{X}$ is the set $\pi _{0}(X)$ of path-connected components of $X$, and $\pi _{\leq 1}\mathbf{X}$ is the usual Poincaré groupoid of $X$.
The next step is to consider an $(\infty ,1)$-category, i.e., an ∞-category whose hom-spaces are ∞-groupoids. This can be thought as a formalization of the naïve idea of having objects, morphisms, homotopies between morphisms, homotopies between homotopies, et cetera. In this sense, endowing a category with a model structure should be thought as a first step towards defining an $(\infty ,1)$-category structure on it.
Turning back to dglas, an easy way to produce nilpotent dglas is the following: pick an arbitrary dgla $\mathfrak{g}$; then, for any differential graded local Artin algebra $A$, take the tensor product $\mathfrak{g}\otimes \mathfrak{m}_{A}$, where $\mathfrak{m}_{A}$ is the maximal ideal of $A$. Since both constructions
and
are functorial, their composition defines a functor
where, by definition, a formal ∞-groupoid is a functor $\mathbf{dgArt}\to \mathbf{\infty \text{-Grpd}}$. Note that $\pi _{\leq 0}(\Def(\mathfrak{g}))$ is the usual set valued deformation functor associated with $\mathfrak{g}$, i.e., the functor
where the gauge equivalence of Maurer-Cartan elements is induced by the gauge action
of $\exp (\mathfrak{g}^{0}\otimes \mathfrak{m}_{A})$ on the subset $\MC(\mathfrak{g}\otimes \mathfrak{m}_{A})$ of $\mathfrak{g}^{1}\otimes \mathfrak{m}_{A}$. However, due to the presence of nontrivial irrelevant stabilizers, the groupoid $\pi _{\leq 1}(\Def(\mathfrak{g}))$ is not equivalent to the action groupoid $\MC(\mathfrak{g}\otimes \mathfrak{m}_{A})\bigl /\bigl /\exp (\mathfrak{g}^{0}\otimes \mathfrak{m}_{A})$, unless $\mathfrak{g}$ is concentrated in nonnegative degrees. We will come back to this later. Also note that the zero in $\mathfrak{g}^{1}\otimes \mathfrak{m}_{A}$ gives a natural distinguished element in $\pi _{\leq 0}(\Def(\mathfrak{g}))$: the isomorphism class of the trivial deformation. Since this marking is natural, we will use the same symbol $\pi _{\leq 0}(\Def(\mathfrak{g}))$ to denote both the set $\pi _{\leq 0}(\Def(\mathfrak{g}))$ and the pointed set $\pi _{\leq 0}(\Def(\mathfrak{g});0)$.
It is important to remark that the functors of the form $\Def(\mathfrak{g})$ are very special ones among all formal ∞-groupoids. To begin with, $\Def(\mathfrak{g})(\mathbb{K})=\{ 0\}$ and so, in particular, $\Def(\mathfrak{g})(\mathbb{K})$ is a homotopically trivial ∞-groupoid. Another characterzing property of the functors of the form $\Def(\mathfrak{g})$ among formal ∞-groupoids is that, under suitable assumptions, they commute with homotopy pullbacks; see Pr10, Lu11 for a precise statement. In other words, if we call “formal moduli problems” those formal ∞-groupoids which satisfy the two conditions we have just observed for $\Def(\mathfrak{g})$, what we are saying is that $\Def$ is actually a functor
And a very good reason for working with ∞-groupoids valued deformation functors rather than with their apparently handier set-valued or groupoid-valued versions is the following remarkable result, which allows one to move homotopy constructions back and forth between dglas and formal moduli problems.
(Pridham-Lurie) The functor $\Def: \mathbf{DGLA} \to \mathbf{\text{Formal moduli problems}}$ is an equivalence of $(\infty ,1)$-categories.
Here the $(\infty ,1)$-category structures involved are the most natural ones, and they are both induced by standard model category structures. Namely, on the category of dglas one takes surjective morphisms as fibrations and quasi-isomorphisms as weak equivalences, just as in the case of differential complexes, whereas the model category structure on the right hand side is induced by the standard model category structure on Kan complexes as a subcategory of simplicial sets. A proof of the above equivalence can be found in Pr10, Lu11.
We will often identify a dgla $\mathfrak{g}$ with the functor $\mathbf{dgArt}\to \mathbf{nilpotent\,\, DGLA}$ it defines by the rule $A\mapsto \mathfrak{g}\otimes \mathfrak{m}_{A}$. With this in mind, we will occasionally apply constructions that generally only make sense for nilpotent dglas (such as $\exp$) to arbitrary dglas. What we mean in these cases is that the construction is applied not to a single dgla, but to the functor from $\mathbf{dgArt}$ to nilpotent dglas it defines. The same kind of consideration applies to our somehow colloquial use of the expression “∞-groupoid” in the following sections; namely, by that we will occasionally mean “formal ∞-groupoid”, or even “formal stack in ∞-groupoids”. The precise meaning to be given to “∞-groupoid” will always be clear from the context.
If $\mathbf{X}$ is a formal moduli problem, then the simplicial set $\mathbf{X}(\mathbb{K}[\epsilon ]/(\epsilon ^{2}))$ has a natural structure of simplicial vector space, and so, via the Dold-Kan correspondence, it is equivalent to the datum of a chain complex: the tangent complex $T\mathbf{X}$ of $\mathbf{X}$. Passing from $\mathbf{X}$ to the associated classical moduli problem $\pi _{\leq 0}\mathbf{X}$, the only datum we read of the tangent complex is its homotopy class, i.e., since we are working on a field, its cohomology. In particular, we have a natural isomorphism
of functors $\mathbf{DGLA}\to \mathbf{\text{Vector spaces}}$ between the tangent space to the classical moduli problem associated to a dgla and the first cohomology group of the dgla seen as a cochain complex. Let us rephrase this in a more explicit form. As we noticed in the previous section, $\pi _{\leq 0}\Def(\mathfrak{g})$ is the functor of Artin rings $A\mapsto \MC(\mathfrak{g}\otimes \mathfrak{m}_{A})\bigl /gauge$, hence
This isomorphism is natural. Namely, given a morphism $\varphi \colon \mathfrak{g}\to \mathfrak{h}$ of dglas, let us write $\Phi$ for the induced morphism of classical moduli problems,
Then the differential of $\Phi$,
is naturally identified with
The second cohomology group $H^{2}$ defines a natural obstruction theory for $\pi _{\leq 0}\Def$, i.e., obstructions for the classical moduli problem $\pi _{\leq 0}\Def(\mathfrak{g})$ are naturally identified with elements in $H^{2}(\mathfrak{g})$, see Ma02. Note that this does not mean that each element in $H^{2}(\mathfrak{g})$ represents an obstruction: one can have dglas with nontrivial $H^{2}$ governing unobstructed deformation problems. The naturally of the obstruction theory given by the second cohomology groups means that, if $\varphi \colon \mathfrak{g}\to \mathfrak{h}$ is a morphism of of dglas, the induced morphism in cohomology,
maps obstructions for the classical moduli problem $\pi _{\leq 0}\Def(\mathfrak{g})$ to obstructions for the classical moduli problem $\pi _{\leq 0}\Def(\mathfrak{h})$. In particular, if the moduli problem $\pi _{\leq 0}\Def(\mathfrak{h})$ is unobstructed (e.g., if the functor $\pi _{\leq 0}\Def(\mathfrak{h})$ is smooth), then
Let $\mathfrak{g}$ and $\mathfrak{h}$ be two dglas. The hom-space $\Hom_{\infty }(\mathfrak{g},\mathfrak{h})$ of morphisms between $\mathfrak{g}$ and $\mathfrak{h}$ in the $(\infty ,1)$-category of dglas is conveniently modelled as the simplicial set $\MC(\underline{\Hom}(\mathfrak{g},\mathfrak{h})\otimes \Omega _{\bullet })$, where $\underline{\Hom}(\mathfrak{g},\mathfrak{h})$ is the Chevalley-Eilenberg-type dgla associated with the pair $(\mathfrak{g},\mathfrak{h})$. It is given as the total dgla of the bigraded dgla
endowed with the Lie bracket
defined by
with $\sigma$ ranging in the set of $(q_{1},q_{2})$-unshuffles and and $\pm$ standing for the Koszul sign, and with the differentials
and
given by
and
An explicit determination for the signs in the above formulas can be found, e.g, in LM95,Sc04. These operations are best seen pictorially:
At least in higher categories folklore. ↩